Structure theorem for mod $p^m$ singular Siegel modular forms

TL;DR

This paper establishes a structure theorem for mod p^m singular Siegel modular forms, showing they are congruent to theta series of quadratic forms with specific level structures, revealing their algebraic and level properties.

Contribution

It provides a new structure theorem linking mod p^m singular forms to theta series of quadratic forms with controlled levels, advancing understanding of their algebraic structure.

Findings

Singular forms are congruent to linear combinations of theta series.

Levels of theta series are of the form p-power times N.

In some cases, levels are exactly p.

Abstract

We prove that all mod $p^m$ singular forms of level $N$, degree $n+r$, and $p$-rank $r$ with $n\ge r$ are congruent mod $p^m$ to linear combinations of theta series of degree $r$ attached to quadratic forms of some level. Moreover, we prove that, the levels of theta series are of the form ``$p\mbox{-power}\times N$''. Additionally, in some cases of mod $p$ singular forms with smallest possible weight, we prove that the levels of theta series should be $p$.